Pi is an infinite, nonrepeating (sic) decimal - meaning that
every possible number combination exists somewhere in pi. Converted
into ASCII text, somewhere in that infinite string if digits is the
name of every person you will ever love, the date, time and manner of
your death, and the answers to all the great questions of the
universe.

12 Answers
12

It is not true that an infinite, non-repeating decimal must contain ‘every possible number combination’. The decimal $0.011000111100000111111\dots$ is an easy counterexample. However, if the decimal expansion of $\pi$ contains every possible finite string of digits, which seems quite likely, then the rest of the statement is indeed correct. Of course, in that case it also contains numerical equivalents of every book that will never be written, among other things.

Why does it seem likely, that the decimal expansion of π contains every possible finite string of digits?
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AlexOct 19 '12 at 9:53

27

@Alex: there's no particular reason for the digits of $\pi$ to have any special pattern to them, so mathematicians expect that the digits of $\pi$ more or less "behave randomly," and a random sequence of digits contains every possible finite string of digits with probability $1$ by Borel's normal number theorem: en.wikipedia.org/wiki/Normal_number#Properties_and_examples
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Qiaochu YuanOct 19 '12 at 16:52

31

@JimboJonny: You’re mistaken. I gave an explicit example of an infinite, non-repeating decimal that doesn’t even contain every digit, let alone every possible finite string of digits. Thomas and antz have also given counterexamples to your claim.
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Brian M. ScottOct 19 '12 at 18:31

Let me summarize the things that have been said which are true and add one more thing.

$\pi$ is not known to have this property, but it is expected to be true.

This property does not follow from the fact that the decimal expansion of $\pi$ is infinite and does not repeat.

The one more thing is the following. The assertion that the answer to every question you could possibly want to ask is contained somewhere in the digits of $\pi$ may be true, but it's useless. Here is a string which may make this point clearer: just string together every possible sentence in English, first by length and then by alphabetical order. The resulting string contains the answer to every question you could possibly want to ask, but

most of what it contains is garbage,

you have no way of knowing what is and isn't garbage a priori, and

the only way to refer to a part of the string that isn't garbage is to describe its position in the string, and the bits required to do this themselves constitute a (terrible) encoding of the string. So finding this location is exactly as hard as finding the string itself (that is, finding the answer to whatever question you wanted to ask).

In other words, a string which contains everything contains nothing. Useful communication is useful because of what it does not contain.

You should keep all of the above in mind and then read Jorge Luis Borges' The Library of Babel. (A library which contains every book contains no books.)

"So finding this location is exactly as hard as finding the string itself" - indeed, rather harder: if I know how long a message is, I have an upper bound on the inormation contained in the encoding. But I have no upper bound on the information needed to represent the index into any given normal number.
–
Charles StewartOct 19 '12 at 12:06

2

What if you layout all the sentences in order of usefulness? :P
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naught101Oct 21 '12 at 5:55

1

@didibus Not really, because natural language satisfies a form of Turing-completeness: you could just say "The answer to problem X is, in binary, one one zero one ..." and proceed to give a binary encoding of a description of your "improved" language followed by an encoding of the message itself. Thus any other Turing-complete language can be delivered in English (and most other natural languages in use).
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Mario CarneiroFeb 5 '14 at 14:13

Another amazing example of how mathematicians use 'normal' to mean 'quite special'.
–
Syd KerckhoveJan 9 at 1:26

2

@SydKerckhove: It's normal in the sense that almost all numbers have this property. Numbers like 7 and 4/3 that lack this property are very rare indeed (though still infinite).
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CharlesJan 30 at 17:21

According to Mathematica, when $\pi$ is expressed in base 128 (whose digits can therefore be interpreted as ASCII characters),

"NO" appears at position 702;

"Yes" appears at position 303351.

Given (following Feynman in his Lectures on Physics) that any question $A$ with possible answer $A'$ (correct or not) can be re-expressed in the form "Is $A'$ a correct answer to $A$?", and that such questions have either "no" or "yes" answers, this proves the second sentence of the claim--and shows just how empty an assertion it is. (As others have remarked, the first sentence--depending on its interpretation--is either wrong or has unknown truth value.)

Special thanks to you sir, for your example !
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WildlingOct 19 '12 at 4:58

1

Is it true 'that any question A with possible answer A′ (correct or not) can be re-expressed in the form "Is A′ a correct answer to A?"' Does that reduce the "all the great questions of the universe" to some inferior subset? I don't know, just asking.
–
LarsHOct 19 '12 at 9:56

4

@LarsH That's a good question--but it starts to push us more into philosophy than mathematics. This re-expression of every great question as a yes-no question requires that you accept that every such question does have a definite answer and that you also accept the Law of the Excluded Middle.
–
whuberOct 19 '12 at 13:40

6

@Lars I was just riffing off Feynman, not quoting him. What he actually said is that there likely is a single equation describing all the laws of physics. As I recall, just collect all the basic equations of physical law (presumably finite in number), express them each in the form $u_i=0$, and then write $\sum_i |u_i|^2 = 0$. This trivial re-expression of things that look complicated into something superficially looking much simpler was my motivation for arguing all the great questions of life can be made into yes-no questions.
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whuberOct 19 '12 at 20:43

Imagine finding your life story: a copiously documented and flawless recounting of every day of your life... right up until yesterday where it states that you died and abruptly reverts back to gibberish. If pi truly contains every possible string, then that story is in there, too. Now, imagine if it said you die tomorrow. Would you believe it, or keep searching for the next copy of your life story?

The problem is that there is no structure to the information. It would take a herculean effort to process all of that data to get to the "correct" section, and immense wisdom to recognize it as correct. So if you were thinking of using pi as an oracle to determine these things, you might as well count every single atom that comprises planet Earth. That should serve as a nice warm up.

I believe the statement could be worded more accurately. Given the reasonable assumption that PI is infinitely non repeating, it doesn't follow that it would actually incude any particular sequence.

Take this thought experiment as an analogy. Imagine you had to sit in a room for all eternity sayings words, without every ever uttering the same word twice. You would very soon find yourself saying very long words. But there's no logical reason why you should have to use up all the possible short words first. In fact you could systematically exclude the words "yes" or every word containing the letter "y", or any other arbitrary subset of the infinite set of possible words.

Same goes for digit sequences in PI. It's highly probably that any conceivable sequence can be found in PI if you calculate for long enough, but it's not guaranteed by the prescribed conditions.

Challenge accepted. In the following file are the first 1,048,576 digits (1 Megabyte) of pi (including the leading 3) converted to ANSI (with assistance from the algorithm described in http://stackoverflow.com/questions/12991606/):

And even if your statement is true with $\pi$, it does not make $\pi$ special. If we hit a real number at random, with probability $1$ we will hit a normal number. That is "almost all" real number is like that. The set of not-normal numbers have Lebesgue measure zero.

Yes and no. Yes, any non-repeating infinite sequence can be translated into an ascii representation of random gibberish, which will of course randomly contain everything. No, that isn't particularly amazing or useful, because whatever message you are looking for is also mistated and refuted an infinite number of times.

(For those that say that it isn't necessarily normal, that is unnecessary as the transformation into ascii can be as complex as you like in order to get the result you desire).